Basics of quintic and quadratic expressions

[runicle]

<b>I need a brain refresher</b> to see if i have everything straight in quintic and quadratic expressions.

The n's in an expression represents how many turns a line would have
The amount of (x+1)^2 means a quadratic curve
The if the n is odd there are no complex roots

Now here is the problem... my textbook doesn't explain in full detail which kind of expressions have how many discrete, complex and equal real roots.

 

[turdferguson]

It helps to visualize these functions. (x^2+1) has no roots over reals, while (x^2-1) has 2 real roots. The degree of the polynomial (Im guessing this is what you mean by n) shows how many potential real roots roots it will have. No matter what, the number of real and imaginary roots will equal n.

The degree can also show how many potential changes in direction there will be. For a parabola (maximum of 2 roots), there can be a maximum of 1 change in direction. These are case specific, though, because x^4 looks exactly like a parabola and only has 1 change in direction. A polynomial of 4th can have a maximum of 4 roots and 3 changes in direction. Compare x^4 with (x+1)x(x-1)(x-2) and (x-2)^4.

This should really be moved to the precalc section